Weighted estimate for the convergence rate of a projection difference scheme for a parabolic equation and its application to the approximation of the initial-data control problem |
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Authors: | A V Razgulin |
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Institution: | 1.Faculty of Computational Mathematics and Cybernetics,Moscow State University,Moscow,Russia |
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Abstract: | A new technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value
problem for a linear parabolic operator-differential equation. The technique is based on discrete analogues of weighted estimates
reflecting the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the right-hand
side, a new convergence rate estimate of order O($
\sqrt \tau
$
\sqrt \tau
+ h) is obtained in a weighted energy norm without making any a priori assumptions on the additional smoothness of weak solutions.
The technique leads to a natural projection difference approximation of the problem of controlling nonsmooth initial data.
The convergence rate estimate obtained for the approximating control problems is of the same order O($
\sqrt \tau
$
\sqrt \tau
+ h) as for the projection difference scheme. |
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Keywords: | |
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